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A236679
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Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.
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20
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1, 1, 1, 1, 1, 3, 4, 2, 1, 1, 3, 13, 20, 14, 1, 6, 37, 138, 277, 273, 143, 39, 7, 1, 1, 6, 75, 505, 2154, 5335, 7855, 6472, 2756, 459, 1, 10, 147, 1547, 10855, 50021, 153311, 311552, 416825, 361426, 200996, 71654, 16419, 2363, 211, 11, 1, 1, 10, 246, 3759, 39926, 291171
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OFFSET
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2,6
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COMMENTS
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Changing the offset from 2 to 1, this is also the sequence: "Triangle read by rows: T(n,k) is the number of nonequivalent ways to place k non-attacking kings on an n X n board." (For if each king is represented by a 2 X 2 tile with the king in the upper left corner, the kings do not attack each other.) For example, with offset 1, T(4,3) = 20 because there are 20 nonequivalent ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other. - Heinrich Ludwig and N. J. A. Sloane, Dec 21 2016
It appears that rows 2n and 2n-1 both contain n^2 + 1 entries. Rotations and reflections of placements are not counted. If they are to be counted, see A193580. - Heinrich Ludwig, Dec 11 2016
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LINKS
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FORMULA
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It appears that:
T(n,0) = 1, n>= 2
T(n,1) = (floor((n-2)/2)+1)*(floor((n-2)/2+2))/2, n >= 2
T(c+2*2,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((2-c-1)/2) + A131941(c+1)*floor((2-c)/2)) + S(c+1,3c+2,3), 0 <= c < 2 where
S(c+1,3c+2,3) =
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EXAMPLE
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T(4,2) = 4 because the number of equivalence classes of ways of placing 2 2 X 2 square tiles in a 4 X 4 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
._______ _______ _______ _______
| . | . | | . |___| | . | | |_______|
|___|___| |___| . | |___|___| | . | . |
| | | |___| | | . | |___|___|
|_______| |_______| |___|___| |_______|
The first 6 rows of T(n,k) are:
.\ k 0 1 2 3 4 5 6 7 8 9
n
2 1 1
3 1 1
4 1 3 4 2 1
5 1 3 13 20 14
6 1 6 37 138 277 273 143 39 7 1
7 1 6 75 505 2154 5335 7855 6472 2756 459
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CROSSREFS
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KEYWORD
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tabf,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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